Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1807 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,32,16,21,14,20)(11,22,24,17,26,35,25,36)(12,27,28,18,30,37,29,38)(13,19,33,40,23,34,31,39);(1,2)(3,7)(4,6)(5,8)(9,11,14,25,16,26,15,24)(10,17,32,35,21,36,20,22)(12,13,28,33,30,23,29,31)(18,19,27,39,38,34,37,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,23)(13,26)(14,28)(15,29)(16,30)(17,34)(19,36)(20,37)(21,38)(22,39)(24,33)(25,31)(27,32)(35,40);(3,8)(5,7)(9,13)(10,19)(11,12)(14,31)(15,33)(16,23)(17,18)(20,39)(21,34)(22,27)(24,28)(25,29)(26,30)(32,40)(35,37)(36,38);(2,6)(5,7)(9,14,16,15)(10,20,21,32)(11,24,26,25)(12,28,30,29)(13,31,23,33)(17,35,36,22)(18,37,38,27)(19,39,34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910290);